The invention disclosed and claimed herein pertains generally to the field of spectral analysis, that is, to the field of determining or estimating the constituent frequency components of a signal occurring over time, and the relative strengths of such frequency components. More particularly, the invention pertains to systems for spectral analysis which employ methods or apparatus structured according to the Fast Fourier Transform technique (FFT). Even more particularly, the invention pertains to such systems wherein the window employed in the Fast Fourier Transformation is iteratively adjusted or adapted in response to the amount of leakage which is present in the frequency domain representations of the signal which are generated by the transformation process.
The Fast Fourier Transform technique has become established as a very useful tool in spectral analysis. By making use of the FFT technique, digital processing technology may be employed to rapidly represent a time domain signal x(t), over a finite time period, by a series of discrete frequencies f.sub.m, each having a frequency domain coefficient X(f.sub.m).
The relationships between time domain signal x(t) and frequency domain coefficients X(f.sub.m) are shown by the following equations, the Discrete Fourier Transform (DFT) pair: ##EQU1## where m and n are both integers, m, n=0, 1, . . . N-1.
To implement the above DFT relationships, x(t) is sampled a total of N times over a time period of time duration T, at sampling intervals of .DELTA.T. Successive samples of x(t) are coupled through an array of N digital filters, each filter corresponding to a frequency f.sub.m and providing one of the frequency domain coefficients X(f.sub.m). Each frequency component is given by the relation f.sub.m =m/T, and each coefficient X(f.sub.m) represents the relative strength of time domain signal x(t) at its frequency component f.sub.m. Alternatively, each coefficient X(f.sub.m) may be thought of us indicating the relative amount of energy of x(t) which is contained in the f.sub.m spectral bin thereof, each spectral bin of x(t) being centered at a frequency f.sub.m and having a very narrow bandwidth.
Each digital filter may comprise an interconnection of physical hardware elements, or alternatively, may comprise a combination of hardware and software elements which are structured to compute one of the frequency domain coefficients X(f.sub.m). In either case, the response of each digital filter has the form sin (Nx)/sin (x). Consequently, each filter has a main lobe centered at its frequency f.sub.m, and has side lobes extending on either side of its main lobe across the frequency range f.sub.o -f.sub.N-1. The side lobes of the filter corresponding to frequency f.sub.m are most significant at frequencies adjacent to f.sub.m in the range f.sub.o -f.sub.N-1.
The response curve of a filter having a given center frequency f.sub.m has a null at each of the other frequencies f.sub.m in the range f.sub.o -f.sub.N-1. Therefore, if a signal x(t) consists only of frequency components f.sub.m, the response X(f.sub.m) of a particular filter will accurately represent the signal strength of x(t) at the frequency f.sub.m to which the particular filter corresponds. However, the representation of signal x(t) as a series of discrete frequencies f.sub.m is an approximation based on processing x(t) over a finite time period T. Consequently, signal x(t) is likely to contain a frequency component f.sub.m '=f.sub.m+.delta., 0&lt;.delta.&lt;1, which is within the bandwidth of the main lobe of a particular digital filter. However, a side lobe of other filters may also respond to f.sub.m ', and it will thereby appear that the spectral bins of x(t) to which such other filters correspond contain spectral energy which they in fact do not contain. This phenomenon is known as leakage. The occurrence of leakage in digital filters structured according to the Discrete Fourier Transform of a signal x(t) is referenced in the literature, for example, in G. D. Bergland, "A Guided Tour of the Fast Fourier Transform", IEEE Spectrum, July 1969, pp. 41-52.
Leakage may pose a serious problem when the FFT technique is employed in spectral analysis. For example, if a signal x(t), having a very faint component f.sub.m, is monitored in a noisy environment, strong noise components contained in the spectral bins of frequencies adjacent to f.sub.m might be sensed by a side lobe of the filter corresponding to f.sub.m, and could mask or prevent detection of the faint component. For example, the f.sub.m spectral bin of x(t) could contain a faint signal component, while noise components could be present which had frequencies in the f.sub.m-1 and f.sub.m+1 spectral bins, which were not precisely centered at frequencies f.sub.m-1 and f.sub.m+1. Consequently, sidelobes of filter f.sub.m would respond to outputs of filters f.sub.m-1 and f.sub.m+1, or in other words, energy from the f.sub.m-1 and f.sub.m+1 spectral bins would "leak" into the f.sub.m spectral bin of signal x(t) and obscure the faint component thereof.
To reduce the effect of leakage, various windowing techniques have been developed to generate a set of adjusted filter outputs X'(f.sub.m), which are intended to represent the respective outputs of the digital filters if no leakage occurs. According to conventional windowing, the output X(f.sub.m) of each digital filter f.sub.m is considered to comprise an initial estimate or approximation X(f.sub.m). The initial estimates of some or all of the filters are weighted by respective predetermined window weighting values, and the weighted estimates are added, the sum thereof comprising an adjusted estimate or frequency domain coefficient X'(f.sub.m). For example, the initial output X(f.sub.m) of filter f.sub.m may be weighted by a factor of 1/2, and the outputs of adjacent filters f.sub.m+1 and f.sub.m-1, X(f.sub.m+1) and X(f.sub.m-1), may be weighted by a factor of 1/4, whereby X'(f.sub.m)=1/2X(f.sub.m)+1/4X(f.sub.m+1)+1/4X(f.sub.m-1). By employing such windowing technique, the contribution to X'(f.sub.m) from frequency components in the f.sub.m+1 and f.sub.m-1 spectral bins is substantially reduced, in relation to the contribution of a frequency component in the f.sub.m spectral bin.
To apply weighting values to the frequency domain coefficients of a signal x(t), the signal may be multiplied in the time domain by a selectively shaped time domain window function. Alternatively, each of the frequency domain coefficients may be convolved with the frequency domain transform of the window function, by means of a convolution circuit.
Conventional windowing techniques have the effect of diminishing the response of the most significant sidelobes of a digital filter f.sub.m. However, such techniques also tend to increase the bandwidth of the main lobe of the filter, whereby the resolution of the filter may be substantially reduced. Also, all of the window weights are fixed, that is, they must be precalculated, preassigned and prestored without regard to the actual spectral energy distribution of a signal received by the system. Therefore, such windowing does not discriminate between sidelobes of the various filters which do contribute to leakage and sidelobes which do not. Such windowing is also unresponsive to the presence or extent of leakage in the output of a given digital filter.